Short Communications

Exit laws and excessive functions for superprocesses

E. B. Dynkin

Cornell University, Department of Mathematics, NY

Abstract: Let $\xi$ be a Markov process with transition function $p(r,x;t,dy)$ and let $X$ be the corresponding Dawson–Watanabe superprocess (i.e., the superprocess with the branching characteristic $\psi(u)=\gamma u^2$). Denote by $\mathcal P$ the transition function of $X$ and put
$$ p_n(r,x;t,dy)=\prod_{i=1}^np(r,x_i;t,dy_i), $$
To every $p_n$-exit law $\ell$ there corresponds a $\mathcal P$-exit law $L_\ell$ such that, for every $t$, $L_\ell^t(\mu)$ is a polynomial of degree $n$ in $\mu$ with the leading term $\langle \ell^t,\mu^n\rangle $. Every polynomial $\mathcal P$-exit law has a unique representation of the form $L_{\ell_1}+\cdots+L_{\ell_n}$, where $\ell_k$ is a $p_k$-exit law.

Keywords: Markov process, Dawson–Watanabe superprocess, polynomial $\mathcal{P}$-exit law.

Received: 27.07.1999

Language: English

DOI: 10.4213/tvp1074

 English version: Theory of Probability and its Applications, 2000, 44:4, 762–767

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